Related papers: Sidorenko's conjecture for blow-ups

For a given graph $H$ and the graphical properties $P_1, P_2,\ldots,P_k$, a graph $H$ is said to be $(V_1, V_2,\ldots,V_k)$-partitionable if there exists a partition of $V(H)$ into $k$-sets $V_1, V_2\ldots,V_k$, such that for each…

An $n$-by-$n$ bipartite graph is $H$-saturated if the addition of any missing edge between its two parts creates a new copy of $H$. In 1964, Erd\H{o}s, Hajnal and Moon made a conjecture on the minimum number of edges in a…

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs…

An influential theorem of Nikiforov states that if an $N$-vertex graph $G$ contains at least $\gamma N^h$ copies of some fixed $h$-vertex graph $H$, then $G$ contains an $H$-blowup of order $c_H(\gamma)\log N$. We provide a new proof of…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D \geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…

Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that…

Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_{\alpha}, V_{\beta}\}$. We prove that for every vertex $v \in V_{\beta}$…

As a directed analog of Sidorenko's conjecture in extremal graph theory, Fox, Himwich, Zhou, and the second author defined an oriented graph $H$ to be tournament Sidorenko (anti-Sidorenko) if the random tournament asymptotically minimizes…

A folklore result on matchings in graphs states that if $G$ is a bipartite graph whose vertex classes $A$ and $B$ each have size $n$, with $\mathrm{deg}(u) \geq a$ for every $u \in A$ and $\mathrm{deg}(v) \geq b$ for every $v \in B$, then…

The famous Erd\H{o}s-S\'os conjecture states that every graph of average degree more than $t-1$ must contain every tree on $t+1$ vertices. In this paper, we study a spectral version of this conjecture. For $n>k$, let $S_{n,k}$ be the join…

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…

For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written…

Let the Andr\'{a}sfai graph $\mathrm{And}_k$ be defined as the graph with vertex set $\{v_0,v_1,...c, v_{3k-2}\}$ and two vertices $v_i$ and $v_j$ being adjacent iff $|i-j| \equiv 1\mod 3$. The graphs $\mathrm{And}_k$ are maximal…

Let $H$ be a graph with $\chi(H) = r+1$. Simonovits's theorem states that, if $H$ is edge-critical, the unique largest $H$-free subgraph of $K_n$ is its largest $r$-partite subgraph, provided that $n$ is sufficiently large. We show that the…

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for…